1. MTH374: Optimization For Master of Mathematics By Dr. M. Fazeel Anwar Assistant Professor Department of Mathematics, CIIT Islamabad 1

2. Lecture 02 2

3. Recap Introduction Why do we need optimization? Some important optimization problems 3

4. Introduction Optimization is the art of obtaining best policies to satisfy certain objectives while at the same time satisfying fixed requirements. Effective decision making both in personal and professional capacity Choosing the best possible option

5. Introduction (Cont…) In this course we will learn how to transform certain practical problems into mathematics. We will also study some mathematical techniques that will help us solve these problems This course will show us how the concepts of calculus, linear algebra and other branches of mathematics can be applied in real life In short we are trying to make the art of effective decision making into a science.

6. Optimization problems Finding the shortest route Finding the quickest route Finding the best route under desired conditions Making an investment decision Maximizing profit Minimizing risk Maximizing stock Designing an airplane with minimum weight and desired capacity.

7. More Optimization problems Best possible utilization of resources Optimize production within the constraints Minimize labor while obtaining desired results Deployment of army to minimize damage by an army attack

8. Today’s Topics Mathematical models Variables and objective functions Some optimization from calculus

9. Mathematical Model A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. Mathematical models are used particularly in the natural sciences and engineering disciplines. They are used in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively. Eykhoff (1974) defined a mathematical model as 'a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form'.

10. Example A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy duty fencing selling for $3 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000?

11. 11

12. Example.

13. Example..

14. 14

15. Summary

16. Thank You